报告题目：An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives
报 告 人：徐岩 教授 中国科学技术大学
报告时间：2020年7月13日 10:00-11:00
报告地点：腾讯会议 ID：197 937 207
会议链接：https://meeting.tencent.com/s/yajaaeliqrhF
校内联系人：陶詹晶 zjtao@jlu.edu.cn
报告摘要：
In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of local discontinuous Galerkin (LDG) method and ultra-weak discontinuous Galerkin (UWDG) method. Firstly, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply the UWDG method to the system. We first consider the fourth order and fifth order nonlinear PDEs in one space dimension, and then extend our method to general high order problems and two space dimensions. The main advantage of our method over the LDG method is that we have introduced fewer auxiliary variables, thereby reducing memory and computational costs. The main advantage of our method over the UWDG method is that no internal penalty terms are necessary in order to ensure stability for both even and odd order PDEs. We prove stability of our method in the general nonlinear case and provide optimal error estimates for linear PDEs for the solution itself as well as for the auxiliary variables approximating its derivatives. A key ingredient in the proof of the error estimates is the construction of the relationship between the derivative and the element interface jump of the numerical solution and the auxiliary variable solution of the solution derivative. With this relationship, we can obtain the optimal error estimates. The theoretical findings are confirmed by numerical experiments.
报告人简介：
徐岩，中国科学技术大学数学科学学院教授。2005年于中国科学技术大学数学系获计算数学博士学位。2005-2007年在荷兰Twente大学从事博士后研究工作。2009年获得德国洪堡基金会的支持在德国Freiburg大学访问工作一年。主要研究领域为高精度数值计算方法。研究工作主要涉及高精度离散格式的设计、分析、及其应用等方面，特别侧重于间断有限元方法及其在流体力学、相场模型、相变问题、水波问题的算法设计、理论分析和应用。2008年度获全国优秀博士学位论文奖，2017年获国家自然科学基金委“优秀青年基金”。徐岩教授入选了教育部新世纪优秀人才计划，主持国家自然科学基金面上项目、德国洪堡基金会研究组合作计划(Research Group Linkage Programme)、霍英东青年教师基础研究课题等科研项目。
